V. Arvind, J. Kobler, and M. Mundhenk. Bounded truth-table and conjunctive reductions to sparse and tally sets. In preparation.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....[31] shows that there is an oracle B such that E B P B O(n) Gammatt (SPARSE) so progress in this direction will require nonrelativizable techniques. There are several open questions involving special reducibilities. We mention just one example. Very recently, Arvind, Kobler, and Mundhenk [2] have proven that P 6= NP = NP 6 P btt (P ctt (SPARSE) where P ctt refers to polynomial time conjunctive reducibility. This strengthens Theorem 1.4. Does the class P btt (P ctt (DENSE c ) have measure 0 in E As noted in the introduction, all known P T hard languages for NP are dense, ....

V. Arvind, J. Kobler, and M. Mundhenk, Bounded truth-table and conjunctive reductions to sparse and tally sets, Technical report, University of Ulm, 1992, Technical Report Ulmer Informatik-Berichte 92--01.

....named above) to some sparse set. Recently, Buhrman, Longpr e, and Spaan [BLS92] have proved the surprising result that SPARSE R p c (TALLY) which implies that R p c (SPARSE) R p c (TALLY) and R p b (R p c (SPARSE) R p b (R p c (TALLY) Using this result, it can be shown [AKM] that if NP R p b (R p c (SPARSE) then P = NP, which unifies Theorem 3.2 with the result of Ogiwara and Watanabe [OW91] We now discuss applications of the above results to the classes UP, PP, and C= P. Since for every set A 2 UP it holds that Left(A) is in UP, it also follows that if UP ....

....Theorem 3.5 If PP is contained in R p bc (R p 1 tt (R p c (SPARSE) then P = PP. By the fact that there exist complete sets in C=P that are one word decreasing self reducible [OL] we can show that if C=P is contained in R p bc (R p 1 tt (R p c (SPARSE) then C=P = P. In [AKM] it is shown that for K 2 fUP; PP; C=Pg, the assumption that K is contained in R p b (R p c (SPARSE) implies P = K. By applying a different proof technique we obtain results similar to Corollary 3.4 for the classes Mod k P, k 2 [CH90,BGH90] Definition 3.6 A set L is perfectly one word ....

V. Arvind, J. Kobler, and M. Mundhenk. Bounded truth-table and conjunctive reductions to sparse and tally sets. In preparation.

.... truth table reduces to a sparse set, then P = NP [OW91] In fact, a more general result holds regarding the impossibility (if P 6= NP) of NPcomplete sets reducing to sparse sets: if any NP complete set bounded truth table reduces to a set that conjunctively reduces to a sparse set, then P = NP [AKM] 1 For the reductions we will discuss, the question of sparse hard sets is equivalent to asking what type of reductions might reduce many one complete sets for NP to some sparse set; we will often use this formulation. 2 A conjunctive reduction from A to B means that there is a Turing ....

....there is a sparse set that is conjunctive hard for NP, then P = NP (Corollary 3.4) In fact, in Corollary 3.5 we establish that if NP R p b (R p c (SPARSE) then P = NP, thus extending the result of Ogiwara and Watanabe [OW91] see Figure 1) Theorems 3.2, 3.8, and 3. 9 originally appeared in [AKM] s s s s s Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma R p b (R p c (SPARSE) R p c (SPARSE) R p m (SPARSE) R p b (SPARSE) R p d (SPARSE) Figure 1: Inclusion structure of some reduction classes to sparse ....

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V. Arvind, J. Kobler, and M. Mundhenk. Bounded truth-table and conjunctive reductions to sparse and tally sets. Ulmer Informatik-Bericht 92-01, Fakultat fur Informatik, Universitat Ulm, Ulm, Germany, 1992.

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V. Arvind, J. Kobler, M. Mundhenk, "Bounded truth-table and conjunctive reductions to sparse and tally sets," Technical Report, Universit at Ulm Fakultat fur Informatik, 92-01 (April, 1992), 1-22.

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